hermitian operators

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QMSE-01 Quantum Mechanics for Scientists & EngineersDavid MillerStanford University

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  • 6.2 Unitary and Hermitian operators

    Slides: Video 6.2.3 Hermitian operators

    Text reference: Quantum Mechanics for Scientists and Engineers

    Section 4.11

  • Unitary and Hermitian operators

    Hermitian operators

    Quantum mechanics for scientists and engineers David Miller

  • Hermitian operators

    A Hermitian operator is equal to its own Hermitian adjoint

    Equivalently it is self-adjoint

    M M

  • Hermitian operators

    In matrix terms, with

    then

    so the Hermiticity implies for all i and jso, also

    the diagonal elements of a Hermitian operator must be real

    11 12 13

    21 22 23

    31 32 33

    M M MM M M

    MM M M

    11 21 31

    12 22 31

    13 23 33

    M M MM M M

    MM M M

    ij jiM M

  • Hermitian operators

    To understand Hermiticity in the most general senseconsider

    for arbitrary and and some operatorWe examine

    Since this is just a number a 1 x 1 matrix

    it is also true that

    g M f

    f g M

    g M f

    g M f g M f

  • Hermitian operators

    We can also analyze using the rule for Hermitian adjoints of products

    So

    Hence, if is Hermitian, with thereforethen

    even if and are not orthogonalThis is the most general statement of Hermiticity

    g M f AB B A g M f g M f

    M M M g M f f M g f g

    M f g f M gf M g

  • Hermitian operators

    In integral form, for functions and the statement can be written

    We can rewrite the right hand side using

    and a simple rearrangement leads to

    which is a common statement of Hermiticity in integral form

    f x g x g M f f M g g x Mf x dx f x Mg x dx

    ab a b g x Mf x dx f x Mg x dx g x Mf x dx Mg x f x dx

  • Bra-ket and integral notations

    Note that in the bra-ket notation the operator can also be considered to operate to the left

    is just as meaningful a statement as and we can group the bra-ket multiplications as we wish

    Conventional operators in the notation used in integration such as a differential operator, d/dx

    do not have any meaning operating to the left so Hermiticity in this notation is the less elegant form

    g A A f

    g A f g A f g A f

    g x Mf x dx Mg x f x dx

  • Reality of eigenvalues

    Suppose is a normalized eigenvector of the Hermitian operator with eigenvalue

    Then, by definition

    Therefore

    But from the Hermiticity of we know

    and hence must be real

    nM n

    n n nM

    n n n n n nM

    M n n n n nM M n

  • Orthogonality of eigenfunctions for different eigenvalues

    Trivially

    By associativity

    Using

    Using Hermiticity

    Using

    and are real numbers

    RearrangingBut and are different, so i.e., orthogonality

    0 m n m nM M

    AB B A 0 m n m nM M 0 m n m nM M

    M M 0 m n m nM M

    n n nM 0 m m n m n n 0 m m n n m n

    0 m n m n 0 m n m n

    m n

  • Degeneracy

    It is quite possible and common in symmetric problems

    to have more than one eigenfunction associated with a given eigenvalue

    This situation is known as degeneracy It is provable that

    the number of such degenerate solutions

    for a given finite eigenvalue is itself finite

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